13/11: Difference between revisions
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== Approximation == | == Approximation == | ||
This interval is well approximated by [[17edo|4\17]] (282.353 cents), and even better, by [[29edo|7\29]] (289.655 cents). | This interval is well approximated by [[17edo|4\17]] (282.353 cents), and even better, by [[29edo|7\29]] (289.655 cents). | ||
{{Interval edo approximation|13/11}} | |||
== See also == | == See also == | ||
* [[22/13]] – its [[octave complement]] | * [[22/13]] – its [[octave complement]] | ||
Revision as of 13:11, 3 November 2025
| Interval information |
neogothic minor third,
major minthmic minor third
[sound info]
In 13-limit just intonation, 13/11 is the tridecimal minor third, neogothic minor third or major minthmic minor third, measuring about 289.2¢. It is the difference between the 11th and 13th harmonics. The (octave-reduced) 11th harmonic (11/8, about 551.3¢) and 13th harmonic (13/8, about 840.5¢) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third – it is a major minthma (352/351) narrower than the Pythagorean minor third (32/27). It can even function as such in a 13-limit neogothic minor triad of 22:26:33, with a 3/2 perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant 16/11 as the outside interval in place of 3/2. The latter triad sounds more like a xenharmonic version of a diminished triad, and could not be confused with simpler diminished triads such as 5:6:7.
13/11 is the classic mediant between the simpler and more familiar ratios 6/5 and 7/6, as it can be given as (6+7)/(5+6). This puts it in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = 20/17, the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = 19/16, the overtone minor third of 19-limit JI, about 297.5¢). (See the diagram below.)
| subminor and minor third | 7/6 266.9¢ |
6/5 315.6¢ | |||||||
|---|---|---|---|---|---|---|---|---|---|
| interval in between | << | 36:35 48.7¢ |
>> | ||||||
| add mediant (13/11) | 7/6 266.9¢ |
13/11 289.2¢ |
6/5 315.6¢ | ||||||
| intervals in between | << | 78:77 22.3¢ |
>> | << | 66:65 26.4¢ |
>> | |||
| add mediants (20/17 and 19/16) | 7/6 266.9¢ |
20/17 281.4¢ |
13/11 289.2¢ |
19/16 297.5¢ |
6/5 315.6¢ | ||||
| intervals in between | << 120:119 >> 14.5¢ |
<< 221:220 >> 7.9¢ |
<< 209:208 >> 8.3¢ |
<< 96:95 >> 18.1¢ |
|||||
Approximation
This interval is well approximated by 4\17 (282.353 cents), and even better, by 7\29 (289.655 cents).
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 4 | 1\4 | 300.00 | +10.79 | +3.60 |
| 8 | 2\8 | 300.00 | +10.79 | +7.19 |
| 17 | 4\17 | 282.35 | -6.86 | -9.71 |
| 21 | 5\21 | 285.71 | -3.50 | -6.12 |
| 25 | 6\25 | 288.00 | -1.21 | -2.52 |
| 29 | 7\29 | 289.66 | +0.45 | +1.08 |
| 33 | 8\33 | 290.91 | +1.70 | +4.67 |
| 37 | 9\37 | 291.89 | +2.68 | +8.27 |
| 46 | 11\46 | 286.96 | -2.25 | -8.64 |
| 50 | 12\50 | 288.00 | -1.21 | -5.04 |
| 54 | 13\54 | 288.89 | -0.32 | -1.44 |
| 58 | 14\58 | 289.66 | +0.45 | +2.15 |
| 62 | 15\62 | 290.32 | +1.11 | +5.75 |
| 66 | 16\66 | 290.91 | +1.70 | +9.35 |
| 75 | 18\75 | 288.00 | -1.21 | -7.56 |
| 79 | 19\79 | 288.61 | -0.60 | -3.96 |
See also
- 22/13 – its octave complement
- 33/26 – its fifth complement
- 44/39 – its fourth complement
- Ed13/11
- Gallery of just intervals
- Gentle chords
- List of root-3rd-P5 triads in JI
- File:Ji-13-11-csound-foscil-220hz.mp3 – another sound example
External links
- The Noble Mediant by Margo Schulter and David Keenan, the earliest description of 13/11 as the "neo-Gothic" minor third